PRACTICAL 




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Book 

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Practical 

Perspective 

▼ ▼ ▼ 

A PRACTICAL EXPLANATION OF THE 

ONLY PRACTICAL PERSPECTIVE 

(ISOMETRIC) 

▼ T T 

By Frank Richards 

Associate Editor "American Machinist" 

and Fred H. Colvin 

▼ ▼ ▼ 

1905 






New York 



365 



LIBRARY of OONGRESS 
Two Copies rtecetvcLi 

JUN 24 1905 

CopyriKnt diirv 
CLASS (/<:^ XAc. Nuj 
COPY s. ' 



Copyright 1905 by 
The Derry-Collard Co. 



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This is to show the ruling of The 

Derry-Collard Co.'s Isometric 

Paper and is not a sample of the 

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The Principles of 
Isometric Perspective 

BY FEAXK EICHAEDS 
Associate Editor " American Machinist " 

T T T 

▼ ▼ 

.1 suppose it is from constantly seeing working draw- 
ings of machinery, and no other representations of it, that 
mechanics generally are entirely unable to sketch anything 
in perspective. Yet, in perspective or pictorial work is 
where free-hand drawing is most valuable to the mechanic. 
Without perspective or without some representations of 
more than one side of an object, a drawing is not readily 
and effectively descriptive to the untrained eye. It seems 
to me that if mechanics could get hold of the simple prin- 
ciples of isometric projection, it would help them much in 
making intelligible sketches of machinery. The mechan- 
ical draftsman does not use it very commonly, and I sup- 
pose the principal reason is the difficulty of drawing 
ellipses. But this objection does not apply to free-hand 
drawing at all, because it is just as easy to draw an ellipse 
by hand as it is a circle, and perhaps a little easier. So 
I will venture to explain the principles of isometric pro- 
jection, so that common folks will not be afraid to use it 
every day, if they choose to. 

The regulation way of teaching the principles of iso- 
metric projection is by the drawing of a cube (Fig. i). 

7 



Practical Perspective. 

That is supposed to show the whole thing. Of course it 
does ; but why not use a brick, that shows it better ? Even 
the word cube is objectionable, because it is one of the 
words that frighten the poor little machinist. But, ser- 
iously, a brick is preferable for our purpose, because its 



Fig. 1 



Fig. 2 



Fig. 3 




Fig. 5 



Fig. 4 



Fig. 7 



three dimensions are all different, and cannot be mistaken 
for each other. 

Now, if we were going to make a working drawing of 
a brick, we couldn't get along without at least three views 
of it : plan, side or front elevation and end elevation. With- 
out all three views we would be ver}^ much in doubt about 
it, and even with all three it would be possible to be mis- 
taken ; so one is really not very much to blame, nor very 

8 



Practical Perspective. 

much to be despised, if he is not quick to see the thing that 
a "draft" tries to represent. With only the plan before us 
(Fig. 2,) the thing represented might be cylindrical or 
round in either direction. With plan and side elevation 
(Fig. 3) it might be elHptical in cross-section. With all 
three views (Fig. 4) it might have the straight lines and 
the square corners at the end shown and the corners might 
taper off and the other end of the object might be oval or 
semi-circular, and the drawing would still be a correct 
representation of it. So to bind the meaning of the draw- 
ing beyond all possibility of mistake, we would require a 
view of the other end of the brick (Fig. .5) or four views 
in all. But with an isometric projection of the brick (Fig. 
6), we would require no label to tell us, ''This is a brick," 
and we wouldn't find it possible to mistake it for anything 
else. By drawing the dotted lines (Fig. 7), the entire 
outline of the brick is shown. This is another advantage 
the isometric projection often gives us. 

It will be remembered that I am calling attention to 
isometrical projection for its value and applicability to 
sketching or free-hand drawing. The little illustrations 
that I offer are made with instruments in the regular 
mechanical way, and that may be said to be rather incon- 
sistent. But I am only trying to put the idea of isomet- 
rical drawing in a simple and familiar way, so that com- 
mon, every-day, young shop folks will be able to get hold 
of it. The application of it will come readily enough to 
any one who once fully understands it ; and it will come in 
play whether making working drawings with instruments 
or sketching with eye and hand unassisted. 

We are now ready to make an isometric projection of 
a brick. Our brick, we need not remind any one, has three 
dimensions : length, width or breadth, and thickness or 



Practical Perspective. 



depth. Either dimension may be assumed for-jjjie vertical 
one. We can draw the brick as standing upon its side 
(Fig. 8), its edge (Fig. 9), or its end (Fig^ "to) ; and so 
the brick can be drawn in isometrical projection in three 
different positions, or, making them right and left, we can 
have six different views, using whichever may be most 
convenient. Strictly speaking, our brick can be drawn in 
any position, as, for instance, where it might occur in the 
setting of a boiler ; but it will be better for us not to say 
anything about that now. 

We can draw our brick first in what we may call its 

Fig. 8 Fig. 10 




fC>\ 




Fig. 11 

most natural position — lying upon its side. Then the lines 
running in the direction of its thickness or height, the cor- 
ner lines, will be vertical. Lines that are vertical in the 
object to be drawn should always be vertical in the draw- 
ing. This law determines the direction of many lines of a 
drawing, without any further trouble, and it may usually 
be well to draw a prominent vertical line of an object for 
a beginning. The character of the object will decide that. 
Having determined the scale that we want to make our 
drawing to, we can draw the first vertical line A B, accord- 
ing to that scale. There is a scale of some kind to every 



10 



^ Practical Perspective. 

drawing. »\\'hen we set out to make a rough sketch of 
anything, wm have some idea of how big we w^ant the 
whole thing to be, and then, from the beginning, we make 
the parts of -it- of a size or length to correspond with our 
idea of the whole, and that is working to a scale in a rough 
and perhaps unconscious way. 

Now when we come to draw our next line we come to 
one of the peculiarities of isometrical drawing, and that is 
that straight horizontal lines in the object that stand 
square with, or at right angles to each other, are drawn 
in one or the other of two fixed directions. We draw 
the line B C for the bottom corner of the front edge of 
the brick. This line should stand at the angle of 60 
degrees from the vertical line, and all lines in the brick 
rimning lengthwise of it, and that are parallel with the one 
represented in the drawing by our line B C, must be drawn 
parallel with that line, or they must all stand at 60 degrees 
from the vertical line. Then the line B D for the bottom 
corner of the end of the brick must be drawn in a direction" 
60 degrees the-other side of the vertical line. In following 
the isometric system for free-hand drawing, it of course 
will not be necessary to determine either of these angles 
with any accuracy. 

We have now the three lines A B, B C, and B D. A 
B is already of the required length according to our scale. 
\A^e now measure off the other two lines to the required 
length according to the same scale, and that brings us to 
the other peculiarity of isometric drawing, the peculiarity 
that is supposed to be described in its Greek and Latin 
name, the peculiarity that makes it peculiarly valuable for 
mechanical drawing from true perspective, all the rectang- 
ular lines, all the lines of a brick, in an isometric drawing 
are of their proper proportional length, according to the 

1 1 



Practical Perspective. 

scale of the drawing. An isometric drawing is thus a cor- 
rect, accurate and reliable working drawing, which a true 
perspective drawing can never be. 

To complete our drawing of a brick no further meas- 
uring will be required. From the point A we draw A E 
parallel to B C, and A I parallel io B D ; then the two ver- 
tical lines / D and E C ; then E H parallel to B D and / H 
parallel to B C finishes the work. 

That is all there is of it, and there is nothing terrible 
about it. Anybody can project a brick. And anybody 
that can project a brick on paper can draw any rectilinear 
rectangular figure. And now, you see, I am in trouble 
again. He will surely take fright at those two terrible 
words. I mean that, as far as we have got now, you can 
draw anything that is made of straight lines that are paral- 
lel to or square with each other. We can now make an 
isometric projection of anything that is made of parallel 
lines and right angles. Where oblique lines and circles 
come into the case we will have a little more trouble, and 
we will not say anything about it now. We will exer- 
cise what we have so as to fasten it. If you can draw one 
brick you can draw several, and I have piled up a few here 
(Fig. ii) that ought to make a good exercise. You can 
find plenty of real models around the shop, and they will 
be of use to you if you use them. 

It will be remembered that I started about this iso- 
metric drawing as useful in the free-hand drawing of 
machinery ; yet I am making so much fuss about it that I 
am afraid I will frighten the sketcher away. But while 
we are on any subject it is always proper to say that the 
more we learn about it, and the more thoroughly we under, 
stand it, the more useful it will be to us. 

The trouble about the isometric business begins when 

12 



Practical Perspective, 

we go to represent anything other than the rectangular 
lines. It is just like it is in life, you know. Anybody that 
is not square is bound to learn what trouble is. Now, 
suppose that we have two bricks, (Fig. 12), one lying flat 
and the other one tipped over against the end of the first 

Fig. 16 Fig. 12 






e g a 



h h 




Fig. U 



Fig. 15 



at an angle of 45 degrees, the front edges of both bricks 
being in the same plane ; that is, if you laid a straightedge 
against the front edge of both bricks, and in more than 
one direction, it would always touch and coincide with 
both. (This is a complete scientific definition, just dis- 
guised enough for me to get it into anybody without 



3 



Practical Perspective. 

frightening them.) Now, none of the lines defining the 
outline of the front edge of the inclined brick, nor those 
at the back edge parallel to them, will be isometric Hnes, 
and we must find some way of getting at both the length 
and direction of them. The remaining lines of the end 
will be isometric lines, and will be parallel to the end lines 
of the other brick. When we get the location of the 
angles from which these end lines start, we can draw 
them to the correct length, according to the scale of the 
drawing. All these drawings of the brick I make to scale, 
and I call the dimensions of the brick 8"x4"x2^", and 
the scale is ^. In Fig. 13 we prolong the base line of the 
under brick as far as seems necessary, and we make the 
isometric drawing of the under brick, Fig. 14, and prolong 
the base line of that, also. Then in Fig 13 we measure off 
the distance, a h, and mark the point, h, in Fig. 14. As 
the angle in this case happens to be 45°, and as the height 
of the under brick is 2^", we know that a b will be 2^". 
Then a line drawn from b and just touching c, will be the 
first line of our other brick ; but we don't know the length 
of it. In Fig. 13 we let fall the perpendicular d e, which 
gives us the distance e a. Then measuring the distance e a 
in Fig. 14, we raise a perpendicular to the height e d 
obtained from Fig. 13. Then d will give us the length of 
d b, which we need not have drawn until now. In the 
same manner we may obtain either the point f or the point 
i, say d f; then the other two sides / i and i b may be drawn 
parallel to d b and d f, and we have the edge complete. / 
k and d I being isometric lines, we may draw them at the 
required angle, and to the length according to our scale, 
and then drawing k I will finish the whole job. 

A peculiar property of the isometric projection is 
brought out by the angle we in this case have assumed for 

14 



Practical Perspective. 

the inclination of the brick. The angle being 45°, but two 
of the six faces of the brick appear. This could only occur 
at this precise angle. If the brick were tipped down more, 
so that the angle with the base line became more acute, 
the upper face of the brick would begin to show, and if the 
brick were tipped up so that the base angle became great- 
er, we would begin to see the under face of the brick. 

Another peculiarity that the use of the angle 45° 
brings to our attention, is the relation of the line f i to k f, 
and of d'b to / d. That they should be in a straight line is, 
indeed, involved in what we were thinking of in the last 
paragraph — the fact that at this angle of inclination the 
upper and under faces of the brick are both invisible, and 
those faces being planes, and coincident with the lines, 
extending to the eye of the observer, the entire visible 
boundary of either face must be one straight line. You 
take anything with a perfectly flat surface, no matter what 
the shape of its outline, rotmd, square, triangular or irreg- 
ular, and tip it up until the face just disappears from sight, 
and all the edge of that surface that you can see will be a 
straight line. But the queer thing about our brick is, that 
one part of the straight line is an isometric line, and anoth- 
er part of the same line is not. f k and d I are true isomet- 
ric lines. Their direction is correct, and their length 
agrees with the scale of the drawing. But / i and d b are 
not isometric lines. Any one can see how they are for 
length, without taking the trouble to measure anything. 
d b should be of the same length as c m, but it is evidently 
shorter than that. So, too, i b should be of the same length 
as c a, but any one can see that it is longer. 

To show what queer capers the isometric drawing 
will cut with an object, I have taken the same brick and 
slid it along and tipped it up against the near end of the 

15 



Practical Perspective. 

under brick, instead of the far end, and at the same angle, 
45°, as before. It is drawn correctly according to the 
same scale as Fig 14. The distortions of dimensions, it 
will be noticed, are here reversed. The brick that was too 
short in Fig. 14, is now too long, and where it was too 
thick in Fig. 14, it is now too thin. In an isometric draw- 
ing, then, to have a line true to scale, it is not enough that 
it should be parallel to one of the three isometric lines. 
We must also know that the line represented stands in the 
object in one of the three rectangular directions that the 
isometric lines represent. We know that the front out- 
lines of our leaning bricks were, at 45°, equidistant from 
the two, and, consequently, as wrong as they could be in 
that plane. It is the same with imaginary lines as with 
actual lines. Let me call attention to the vertical dotted 
line marked x. It is of the same length in all the figures, 
and in the isometric projections it is a true isometric 
line, and is correct to scale in length, except that in the 
thin brick, Fig. 16, it is apparently a little longer than in 
the thick brick. Fig. 14 ; but that is only because I am not 
the most accurate draftsman in the world. 

It is proper, also, to call attention to the angles in iso- 
metric drawing. As a general rule, we may say that they 
are always wrong. They are not, in the isometric draw- 
ing, the same as in the object represented. We have been 
representing our brick tipped over at 45° ; but that is not 
the angle in the drawing. And, curiously enough, the 
corners of the brick itself, that is tipped over at 45°, are 
correct. The brick being wrong, its angles are right. But 
in the under brick, which we assume to lie on level ground, 
none of its angles are right angles. If we ever become 
expert enough in isometric drawing to dare to tackle cir- 

16 



Practical Perspective. 

cular work, we may learn more of how angles are squashed 
and jammed out of shape by isometric projection. 

From all these things we may infer that an isometric 
drawing is fearfully and wonderfulty made. It is not a 
thing to be trifled with. As a working drawing in a shop 
it wouldn't do for an ignoramus to try to measure it any- 
where. For the working measures of parts he must follow 
implicitly those that have been written in by competent 
and responsible persons, and stray not either to the right 
or to the left. I have no expectation that isometric draw- 





rig. 18 



Fig. 17 

ing will be generally adopted for working drawings, and 
I do not advocate it ; but I do k»iow that it would enable 
many a workman to get an idea of the shape of a piece, 
and of how the different pieces of a machine are related. 
Like everything else, it is valuable in its place, and would 
be more generally used if mechanics, and especially young 
mechanics, would take the trouble to get the hang of it. 

Generally, in drawing machines or parts of machines 
in isometric projection, no plane projection, for the pur- 
pose of getting construction lines, or of locating points, 



17 



Practical Perspective. 

will be needed, as there will be rectangular lines in the 
thing represented, with which any oblique lines that occur 
will be in contact, or to which they can be referred. 

Here is a horse, a familiar quadruped of the shop, 
(Fig. 17). To draw it nothing is needed but the single pro- 
jection and such construction lines as we can readily locate 
by it. We need the measures of the parts, of course, but 
those we need in any case. The body of the horse is rect- 
angular, and we draw the entire outline of it according to 
our scale, the parts out of sight being in pencil in our 
drawing, and shown here by dotted lines. 

It occurs to me, just here, that we ought to have a 
name, as a labor-saver, by which to designate the three 
normal isometric lines. A, B, C, Fig 18. The vertical is, 
of course, self-named. The other two we may call the 
right isometric line, A, and the left isometric line, C ; and 
so I use them. 

Well, from the center of the near end of the body of 
the horse we drop the vertical line a h, the length being 
equal to the height of the horse. Then from h we draw 
the right isometric line h c, which will be the center line of 
the base. Along this center line we measure and mark the 
four points e, i, 0, 11, corresponding with the same points 
upon the top of the horse. Through each of these we draw 
a left isometric line, and*>space one of them equidistantly 
upon both sides of the center line, the sum of the two 
measures being equal to the spread of the legs at the base. 
Then by two right isometric lines we carry these same 
measures of the first left isometric line to the other three 
of them. The vertical lines at the sides of the body where 
the legs meet it are then to be indicated. We have now all 
the marks necessary to complete the horse, which we pro- 
ceed to do by drawing the remaining lines. The outer cor- 



Practical Perspective. 



ner lines of the legs being drawn, the inner lines are simp- 
ly drawn parallel to them, and meeting the body at the 
under corner of it. 

The phenomenon regarding the change of oblique 
dimensions is again here to be observed. The length of the 
front legs is too long, and the length of the back legs is 
too short. This is not at all due to the distance that they 
stand from the eye, as would be the case in true perspec- 
tive, but is entirely on account of the different angles at 
which the legs stand. There seems to be a law of com- 
pensation in the matter, too, for, where the leg is elongated 

Fig. 20 




Fig. 19 



it also becomes thinner, and where it is shortened it be- 
comes thicker. 

Here are four bricks, Fig. 19, placed centrally and 
rectangularly, and at the same angle of inclination upon 
the two edges and the two ends of a brick lying flat upon 
its side. This is an easy thing to draw, the only assistance 
required being one elevation. Fig. 20, of the reclining 
brick, from which all necessary measures may be taken. 
All four bricks reclining at the same angle, the bottom 
corner of each is at the same distance from the under 
brick, and the upper corners are all of the same height. 
Fig. 19 shows rather strikingly, T think, how tlie isometric 
projection may change the appearance of an object. The 

19 



Practical Perspective. 

bricks at the back would scarcely be accepted by the unin- 
itiated as identical in size and position with those in front. 

In practical drawing, the same as in machine work, 
in the preliminary construction of a machine upon paper, 
as in the actual work of the shop, the faculty of invention 
is always in play. Different ways of doing things are 
always to be thought of, for it is seldom that the best is 
thought of first. As I believe in wringing things as dry 
as possible, as I always advocate the getting as much as 
possible out of everything before leaving it, I may be per- 
mitted to follow the practice that I advocate, and bring up 
what we have already mentioned before. In the drawing of 
the brick, (Fig. i6), inclined at 45°, instead of drawing 
any of the dotted construction lines, as we did before, ex- 
cept the base Hne, we could have got all the required points 
by arithmetic ; by the arithmetic of the common school that 
the mechanic should carry into the shop with him when he 
enters it. I always say, as I always believe, that arithmetic 
is as valuable to the mechanic as any tool he can have, and, 
like all his tools, it is not of much use to him, and does not 
speak well of him if it is allowed to get rusty. So it will 
be good for it and for us if we brighten it up a little by 
exercising it in this case. This will give us quite a neat 
little job in arithmetic, bringing into play the three most 
important arithmetical operations the mechanic has to use : 
the extraction of the square root, finding the third side of 
a rightangled triangle, and simple proportion. I would 
Hke to say more about machine shop arithmetic by and by, 
if we travel together so far, but just now we will stick to 
the brick case. 

The angle of inclination, Fig. 21, being 45°, b c must 
be equal to a b, and in Fig. 22 we can measure off b c and 
then draw c a, and prolong it upwards beyond <7. as far as 

20 



Practical Perspective. 

may seem necessary. This will be the lower line of the 
inclined brick, but not being an isometric line we cannot 
measure off the length of the brick by our scale. We first 
find the length of a c in Fig. 21. This is the hypothenuse 
of the right-angled triangle a b c, so we get 2.5^+2.5-= 
12.50, and V 12.50=3.5355, the length of a c. Then the 
length of the remainder of the brick will be 8 — 3-5355= 
4.4645. Now this, instead of being the end, is only the 
beginning. The length a c m Fig. 21 is correct accord- 
ing to our scale, but in Fig. 22, while a c represents the 
same portion of the total length of the brick as a c in Fig. 
21, its actual length is not the same. It is evidently longer, 
and we want to find what its actual length is. To show 
this comfortably without hitting anything, we draw this 
on a larger scale. Fig. 23. Now, from c as a center, with 
the radius c h, we draw the arc bid and the line c d, mak- 
ing the angle i c d 30°, the same as we know that the 
angle i c b is. If we have been studying our geometry, as 
I assume that we have been, we know that b d is, equal to 
c b, and consequently e b is equal to one-half of c b, or 
1.25. Then from the right-angled triangle bee we get 
2.5- — 1.252=14.6875 and V 4.6875=2.16, the length of e e, 
and from the right-angled triangle a e c we get (2.5 + 
i.25)-+2.i6-=i8.728i, and V 18.7281=4.32, the actual 
length of a c. But this actual length (by the scale of the 
drawing) of a c represents only 3.5355, and if 4.32 repre- 
sents 3.5355, what will represent 8, the total length of the 
brick ? This we find by the simple proportion : 3.5355 :8 : : 
4.32:? and solving this we have 8X4.32-=-3.5355=9.77. 
This 977 then will be the length c a u in Fig. 22. 

To complete the brick we have now to locate the point 
0, after which the remainder of the figure is determined 
by lines drawn parallel to those already made. In Fig 21 



Practical Perspective. 

X c being a right-angled triangle, and the angle o c x 
being, as we know, 45°, and the side c opposite the right 

angle being 2.5, we get y — = 1.7, which will be the 

2 
length of both x and c x. These lines will be, in Fig. 22 
or 23, true isometric lines, and drawing them according 




Fig. 23 

to our scale gives us the point 0. From this we draw g 
to scale, and then the parallel lines that finish the brick. 

We may now venture to look a little at the standard 
and traditional diagram that has always been used to show 
the principles of isometric projection. It is very conven- 
ient for the purpose, and I have not meant to treat it with 
any disrespect. Here is a plane projection of a cube, Fig. 
24. Of course all its sides are equal, and all its angles 
are right angles. Here is the isometric projection of a 
cube, drawn to the same scale, Fig. 25. The sides are still 



Practical Perspective. 

equal to those in the plane projection, and still equal to 
each other; but the angles are changed. Two opposite 
angles of each face of the cube have become obtuse angles, 
and the two alternate opposite angles have become acute 
angles. The square has become a rhombus. 

Now no mechanic has any respect for a rhombus. He 
always thinks of it as a square that has been spoiled in 
making. It has tipped over, (Fig. 26), before its joints have 
set, you know. But the same mechanic has great admira- 





Fig. 24 



tion for the diamond ; so here we present him with a cou- 
ple of them, Figs. 2^] and 28. You will have hard work to 
persuade him that they are the same old rhombus in dis- 
guise. It may be that the disguise was the other way, and 
that what we habitually think of as the diamond is the real 
thing. We certainly know more of its properties under 
that name. We regard it as one of the most symmetrical 
of figures. It is symmetrical in relation to both of its axes 
or center lines. When a square becomes a rhombus, as 
it does when we make an isometric projection of a square 



23 



Practical Perspective. 

surface, the thing that occurs in connection with the 
change of the angles is a change in the length of the diag- 
onals, or the lines connecting the opposite corners of the 
square. One becomes longer and the other becomes short- 
er. But, however much the square may be tipped over, 
and however much the angles may be changed, the two 
diagonals always remain perpendicular to each other. We 
encounter this principle frequently in machinery, in "lazy 
tongs" and movements of that character. This was why 
the corners of our brick, tipped over at 45°, appeared as 
right angles. The sides and ends of the brick had then 
come into the position of the diagonals of a square. 





Fig. 26 



Fig. 27 



Fig. 28 



The rhombus that occurs in true isometric projection 
always has the same angles, so we need not chase our dia- 
mond to any extremes. In isometric drawing the propor- 
tional length of either diagonal of the rhombus to the side 
of it is always the same, and the proportional measure of 
any figure, when measured in a direction parallel to the 
diagonal, will be changed in the same ratio. A circle lying 
in the plane of either square face will be elongated in the 
line of the longer diagonal, and in the same proportion as 
the diagonal is itself elongated, and it will be shortened in 
the line of the shorter diagonal. The circle becomes an 
ellipse, and it is an ellipse of constant shape. If we can 



24 



Practical Perspective. 

find the constant relative lengths of the two diagonals we 
can know the constant ratio of the two axes of the ellipse 
to each other, and the ratio of each to the actual diameter 
of the circle represented. 

The diagonal a & of the square, Fig. 24, being the 
hypothenuse of a right-angled triangle, and the other two 
sides each being i, the diagonal will be V 2, or 1.414. In 
the isometric drawing of the square, the right isometric or 
base line b c being 30° from the horizontal, and a h being 
vertical, the angle a h c must be 60°. Now we know, 
because everybody knows, or ought to, that 60° is the 
angle of an equilateral triangle, and the triangle ah c must 
be such a triangle. Then a c must be the same length as 
the other sides of the triangle. Its length must be i. But 
its original length in the square was 1.4 14, and its present 
length is a fraction of its original length represented by 

I 

, and the measure of everything on this line will be 

1.414 

this fraction of its original length. To obtain the length 
of the long diagonal we have here another right-angled 
triangle a d c, that will readily give us half of it, or the 
same triangle would do it. The length oi d c \s i, and e c 
of course .5, then the length of d ^ is V i^ — .5^=.866. 
This being one-half the long diagonal, its whole length 
must be 1.732. Its present relative length, then, treated 

1.732 
the same as we served the short one, will be . If now 

1.414 
we make both of these into decimals, by dividing the num- 
erator of each by its denominator, we get .707 and 1.224 
as the proportional (not actual) lengths of the diagonals, 
and the proportional lengths of the axes of the ellipse will 

25 



Practical Perspective. 

be the same. These will also be the actual lengths of the 
axes where the original diameter was i. 

As I believe I said in the beginning, the difficulty of 
readily describing ellipses has had much to do with pre- 
venting the general use of isometric projection. If an 
ellipse could be described as readily as the circle can be 
drawn with compasses, we would use it more, and we 
would know more about it. I remember that I learned 
much of the properties of circles by scratching them on 
the ground with the tines of a pitchfork. An ellipsograph 
is too expensive a plaything for every boy to have; and 
the best of them, so far as I know, have the fatal defect of 
not making a good line in ink. They will describe an 
ellipse very nicely with a pencil, but, after all, the ellipse 
must be inked by hand, so that what we use to ultimately 
guide the pen in inking, we use for the whole process, and 
we put the ellipsograph carefully away upon the top shelf. 

The ellipses made in sets that I find in catalogues of 
drafting instruments are not of the right proportions for 
isometric drawing. In those that I have examined — and 
I know of no others — the short diameter is .75 of the long 
diameter. The short diameter, the long diameter being t, 

.707 

of the isometric ellipse, is : = .579. The scale which 

1.224 
I offer. Fig. 29, half size, should be of some use. This 
will readily show the semi-diameters of any ellipse within 
the range of it. The base line represents the actual diam- 
eter of the circle, and upon it may be raised vertical lines 
representing as many subdivisions of the inch as may be 
desired. Where these verticals are cut by the two oblique 
lines will be the corresponding semi-diameters of ihe 
ellipse. 

26 



Practical Perspective. 

The ellipse, Fig, 30, will touch the sides of the rhom- 
bus, as the circle touches the sides of the square, at the 
middle of each. Then if upon the two diagonals we meas- 
ure off the extreme dimensions of the ellipse, as obtained 




Fig. 29 




from the scale, Fig. 29, or otherwise, we have eight points 
■of the ellipse, and we can readily draw it in with sweeps. 

If eight points are not thought to be enough to define 
the ellipse accurately, we can readily obtain as many as we 

27 



Practical Perspective. 

choose. One method would be by the subdivision of the 
square into any number of smaller squares, and a corre- 
sponding sub-division of the rhombus. This would apply 
to the reproduction of any figure, regular or irregular, in 
any isometric plane. The five-pointed star, Figs, 31 and 




Fig. 31 



Fig. 32 



32, is not difficult, because only the five extreme points 
have to be located, the straight lines determining all the 
rest, and the system of squares was not necessary. By the 
way, Fig. 30, by turning it around, will correctly represent 
either face of the cube. 



28 



The Use of . 
Isometric Paper 



BY FEED H. COLVIN 



T ▼ T 
T T 



Mr. Richards has explained the principles of isomet- 
ric perspective, which is the only practical perspective for 
the mechanic, and has shown some of its applications to 
machine work. That it has not become much more com- 
mon can only be due to the fact that it has not been under- 
stood, that it was too much trouble to get at it and the 
trouble of laying it out on ordinary paper. 

By the use of the D-C Isometric Sketching Paper 
nine-tenths of the difficulties disappear and a sketch can be 
made in short order without the use of anything but a 
pencil as even a rule or a 30 degree angle is not needed. 
If a finished drawing is desired you only need a compass 
and a rule in addition to the ruling pen as the sectioning 
of the paper takes care of all the rest. 

After what has gone before it will only be necessary to 
give a few suggestions and show their actual application 
by examples. Free hand sketches are perhaps the most 
useful in nearly all cases and can be made readily, but if 
correctness and finished drawings are desired they can 
easily be obtained. 

One of the first things to get straight in your mind 
is the proper position of the ellipses which represent the 
circles. Unless this is done some queer errors occur, but 
a few moments will make it so clear that you will never 

29 



The Use of Isometric Paper. 

make a mistake in the matter. To show this clearly we 
will take up the old familiar cube which becomes a hexa- 
gon in isometric perspective as shown in Fig. 33, and we 
can make this clear. 

Each side of the cube becomes a diamond as will be 
seen, and the ruling of the paper which is shown to make 
things clear, shows you exactly how these diamonds are 
located in each case. Each side of the cube is nine spaces 
long, and remembering that the dimensions are only to be 
taken along the isometric lines we have no trouble in 
laying off all three sides of the cube or anything else. The 
ruling forms these diamonds in the three directions re- 
quired for showing square or round surfaces in isometric 
perspective as we shall see in every case. These diamonds 
are drawn a little more lightly than the ellipses which it 
is especially intended to illustrate. 

As each side of the cube is nine spaces long the 
diameter along the isometric lines a.t A to B in all three 
cases will give the true diameter. This gives us four 
points in the ellipses, or in other words the center of each 
side of the diamond gives a point. To obtain the other 
four points, the long and short diameters there are several 
methods, two of which are shown here. 

A circle struck from O as a center and touching the 
sides of the hexagon gives very nearly the outer curve of 
the ellipse in all three diamonds and also locates the outer 
side of the short diameter. The same radius from the 
outer corner will give the other side. 

The top ellipse shows one way of finding the other 
diameter, which is easy but not quite correct enough for 
use if you want to lay out work from it. This is to draAv 
a line from the middle of the side of the diamond or from 
the point already found, and use the point where this 

30 



The Use of Isometric Paper. 




31 



The Use of Isometric Paper. 

crosses the center line as a center for the arc. This gives 
too large a curve and too short an ellipse, as shown by the 
dotted lines at the end. 

A much better way is to draw a circle representing 
the small diameter, as shown at the right, and where this 
cuts the center line is the center for the end curve. This 
construction, which is better in every way, involves the 
use of the diagram in Fig. 34, which explains itself and 
which can be used more easily than any set of curves. For 
most work you do not need extreme accuracy and you 
soon get to know the right proportion by noting the dia- 
monds around or in which it is drawn ; for it will usually 
come right to use them. The diagram just referred to 
also gives the correct radius for the larger and smaller 
curves and will be a great time saver where accuracy is 
required. The correct outer curve is not exactly as 
appears in this figure but very near to it. The diagram, 
(Fig. 34), is not claimed to be new with the exception of 
giving proper radius for each curve. 

There is one vital point to be remembered regarding 
these ellipses and this is which to use to represent the 
view you desire, but this is easily learned. Whenever it is a 
top view or shows a circle on the top of any flat surface, 
the ellipse is always as shown on top of the cube or in the 
horizontal diamond. Don't try to twist it around because 
the rest of the piece is off at some angle or other for it 
belongs just where it is shown and will look all right when 
you get it done. If the rod or shaft or pipe runs up and to 
to the left, the ellipse, showing the end of said rod, 
shaft or pipe, will always be in the right hand dia- 
mond and the opposite is true if it runs up and to the right. 
If it runs down this is reversed. Remember this and you'll 
have no trouble on this score. On the paper itself the 

32 



The Use of Isometric Paper. 




33 



The Use of Isometric Paper. 

ruling is very lightly printed so as not to be in the way, 
but the diamonds are there and they will help you draw 
ellipses easily and rapidly, either free-hand or with 
instruments. 

One more elementary lesson and we will get down to 
real drawing of things we see every day. To still further 
show how the ellipse affects things in isometric drawing 
and the value of these ellipses to give us the proper dimen- 
sions of other parts, we show a box in Fig. 35 with 
covers both top and bottom. This practically explains 
itself, as it is easily seen that the length of the sides of the 
cover is always determined by the ellipse. In the position 
indicated by the long diameter the cover appears longer 
than it really is, as the correct dimension is along the 
isometric lines shown by the fully drawn cover or the 
sides of the box. This will be found useful in many cases 
and can be referred to for guidance at any time. This has 
a close relation to the drawing of hexagon and other nuts, 
and it may be well to get these straight before we start to 
draw any machine details. 

Fig. 36 shows a cube with regular hex nuts on the 
two sides and a hex flanged nut on top. The flange is 
simply two ellipses drawn one above the other and the 
hexagon is located in its proper place above. This is one 
of the puzzling things at first. This will be cleared up a 
little later and the ruling of the paper makes it easy after 
we understand what is wanted. 

Fig. 37 shows the corner of a cube and the ellipses 
laid out on the three surfaces. Hexagons or squares are 
laid out practically as in any drawing except that we must 
only consider the center lines and remember that these are 
always at the isometric angle. Taking the ellipse on top 
we sketch in the diameters A, B and C, D. Then for con- 

34 



The Use of Isometric Paper. 




35 



The Use of Isometric Paper. 




36 



The Use of Isometric Paper. 

venience sketch in the Hnes E F, G H, I J, and K L. 
Where these cut the ellipse give the points for the two 
hexagons thrown into the right perspective. Connect 
A E G B H F and A as shown in solid lines and you have 
the hexagon whose true long diameter "across the corners'' 
is A 'B. Connecting I C J L D K and / as with the dotted 
lines gives the hexagon turned so as to bring the true 
long diameter C D. After we get the idea it will be un- 
necessary to draw the construction lines at all, but we can 
just follow the ruling of the paper and locate all the points 
in less time than it takes to tell of it here. For completing 
the nut we simply erect vertical lines from the six points 
to the required thickness of the nut and connect them at 
the top and you have your nut complete. If it is a nut 
with a rounded edge it W\\\ appear as shown on sides of 
Fig. 36. 

The same method will lay out the hexagon nut for 
you in any position bearing in mind the same proportions 
as you always use in laying out a hexagon nut, i. e. with 
the flat side towards you; this flat side is as long as the 
other two which you see. A square nut is shown on the 
left side of Fig. 37, and with this in mind there will be no 
need to spend further time on this detail. If it were not 
for the ruled lines on this paper it would be necessary to 
spend much more time on it and might be well to give a 
table of proportions for the different styles, but the ruling 
makes all this superfluous and as will be seen makes it easy 
for anything to be laid out after one understands the 
principles of isometric as laid down b}' Mr. Richards. 

In Fig. 38 we try our hand at a sketch of what might 
be called a square crank. Taking A B as the diameter 
across the corners of the shaft we sketch up to C D where 
the crank starts oft" to the right. Complete the diamonds 

37 



The Use of Isometric Paper. 




38 



The Use of Isometric Paper. 




39 



The Use of Isometric Paper. 

either in imagination or by dotted Hnes as a guide. Lay- 
ing off the offset of the crank X we spot out the points E F 
and sketch up to the other side of the crank at G H a.t d. 
distance Z from the inside face of the lower part of the 
crank. Setting out points I J we have the continuation 
of the main shaft and can then look after the crank 
cheeks. These can be made perfectly plain or can be offset 
as shown, but there will be no difficulty in laying them off 
right and quickly if we get started right with a few guide 
points as shown, and after one gets the run of this paper 
it can be done without a moment's hesitation. To make 
this regular crank shaft all that is necessary is to sketch 
ellipses over the diamonds and the square shafts become 
round. This is given simply as an example in laying out 
the different points so as to get things just right. 

Going to Fig. 39 we have a genuine crank shaft of 
the engine variety. The first thing to do is to lay the 
center lines for the crankpins, the line from this to the 
center of the shaft and the center line of the shaft. These 
are A B, C D and E F. The center line of the ellipse, 
G H, locates the outer end of the collar on the crank and 
/ / for the inside of this collar. The distance between 
these lines is the thickness of the collar. From I to K 
is the length of the crank pin, and from K to ]\I the 
height the hub extends above the web of the crank. The 
ellipse around K L gives the face of crank hub or boss, 
and around M N (or the portion which would show) the 
corner in the crank between web and hub. Center lines, 
O P and K L both cross their center lines along the line 
C D, and the distance between A and E is throw of the 
crank. Distance between and Q is the thickness of the 
main hub or crank, and the ellipses complete the sketch. A 
little practice will enable anyone to make anything of this 

40 



The Use of Isometric Paper. 




The Use of Isometric Paper. 

kind that may come up in regular work, in a very short 
time, and there isn't a man in the shop who cannot see at 
a glance just what is wanted much better than with the 
regular three view sketch. 

Fig. 40 shows a three way elbow with a piece of pipe 
in one end and illustrates the position of the ellipses for 
all three directions as well as any simple everyday object 
that we can find. 

Fig. 41 shows the business end of a connecting rod of 
a locomotive or other engine. The only things that will 
bother at all in doing this is the offset of the rod brasses 
which projects outside the strap and the set screw and 
key. The oil cup is easy when we remember that the 
ellipse is horizontal in all cases of this kind. As a guide 
in seeing just how this was laid out we have put a few 
construction lines which, however, were not used in 
making the drawing as they are not necessary after one 
has a little experience with the use of the paper. The 
dotted lines showing the hidden parts of the key and oil 
cup were put in to make clear just how the different 
points were obtained and to show that they are located 
correctly. 

Figs. 42 and 43 show wrenches for shop use. The 
monkey wrench is of a knife handle variety and not drawn 
to any special scale but to show that it can be readily 
handled in this way. The dotted lines at the head clear up 
any confusion as to proper location of lines there, while 
method of locating the rivet in the handle is shown by 
the dotted lines there. The distance from the center line 
to the center line of the rivet is half the thickness of the 
handle. 

In the special wrench below we have a square opening 
socket wrench with a piece of pipe through to four hole 

42 



The Use of Isometric Paper. 




43 



The Use of Isometric Paper. 

opening in the handle. This would be rather a difficult 
thing to show so a workman who was not familiar with 
drawing could readily understand if we used the regular 
three view projection method, but there is not a man who 
calls himself a machinist who would hesitate a minute on 
such a sketch as this. You can dimension every part that 
you need to and give it to the mechanic without any fear 
of his going astray on the job. In fact this is the really 
useful field for this method of drawing and the paper here 
described. It gives a means of making rapid and accurate 
sketches freehand that can be given to any workman with 
a knowledge that he can understand it. If finished draw- 
ings are wanted they can be made as we have seen, but 
one of the most useful fields is the drawing room sketch. 
This can either be copied in a press or blue printed as 
desired. 

The next sheet shows a number of rough free hand 
sketches. Needless to say they were not made by an 
expert and can be equalled by any one in the drawing 
room and vastly improved upon by all who are at all 
expert in free hand work. 

The regulation two- jawed chuck in Fig. 44, the core- 
box in 45, the end of an / beam with a few details of 
connections, all explain themselves. Just try to sketch out 
the pipe connections shown in Fig. 47 by the three view 
process and see if you could understand it yourself if it 
was sprung on you suddenly. Would any plumber or 
steam fitter have any trouble in knowing just what you 
wanted from the sketch shown ? And this could have been 
elaborated almost without limit and yet be clear because 
you can see everything and put down any dimension you 
want to. 

Fig. 48 might be a sample block of joints for a 

44 



The Use of Isometric Paper. 




45 



The Use of Isometric Paper. 

manual training school or a key piece to a puzzle, but 
whatever it was there would be no trouble in any carpenter 
or machinist making a piece like it out of either wood or 
iron if the dimensions were given, as might easily be done. 

A blacksmith would welcome a sketch like Fig. 49, 
for he could see exactly what he wanted. These might be 
multiplied many times, but these are sufficient to show 
what can be done with a very little practice. 

The three following figures show details of different 
kinds of machine construction which do not need to be 
described in detail, and the ruling Hues are given to show 
how easily the work may be done. In fact the work can, 
for the most part be done without even the aid of a rule. 
Simply call each space between lines an eighth of an inch, 
if it is small work, an inch or a yard, so long as it is always 
the same and you can lay out anything you want with only 
a pencil and a rubber. The latter not to hide mistakes but 
to erase construction lines which often make the work 
easier. 

Fig. 53 shows the assembly and details of a head- 
stock. There is no need of detailed explanation as any 
mechanic can see just what it is. 

Taking Fig. 54 as an example and we find a piece of 
structural steel construction work which is surely clear for 
any one. The shapes used and the methods of joining 
are made very clear in a sketch like this and men in the 
field would have no hesitation in following the drawings 
without thought of a mistake. 

Beginning at the bottom plate we outline it, go up a 
half space for its thickness and another half space for the 
lower angle used as the fastening. Locate where the Z 
bars will come by means of the ruling and lay out their 
shape at the top of the sketch with the plate between them 

46 



The Use of Isometric Paper. 




47 



The Use of Isometric Paper. 




48 



The Use of Isometric Paper. 




49 



The Use of Isometric Paper. 

to form a web as shown. Then the box beam, made up of 
two channels and two plates is located in between the Z 
bars and the supporting angle put underneath. The ruling 
of the paper takes away all the necessity for T squares or 
angles and you can lay off distance or offsets by simply 
following the lines in the right direction. This will some- 
times be up and across to get the right offset, but you can 
always tell when you have it right or wrong, as it shows 
at a glance when you get a little used to it. Everything 
that is square or rectangular will come out exactly in dia- 
monds every time and if it doesn't you will find some point 
which isn't quite exactly right. 

Architectural details can also be very nicely handled 
in this way, even complete drawings as will be seen from 
some of the illustrations given further along in the book 
to show what has been done with it. 

Fig. 55 shows the corner of a building with the 
cornice and its details all drawn free hand. These ellipses 
are oposite from any others we have used as their lines of 
perspective run down in each case. If you ever have any 
doubts as to the position, sketch ona in lightly and you 
can tell at a glance whether it is right or wrong. 

Fig. 56 is from a set of concrete steps and railings 
from recent practice in railroad station work, and simply 
shows another aplication of this method of projection. 

The illustrations which follow are all drawn in this 
perspective, some with this paper and some without it, 
although the ruling is omitted to show how they look with- 
out it and also how much more difficult it is to draw with 
the specially ruled paper. As each illustration carries its 
own explanation there is no need of further description in 
the text. With these examples, which include nearly all 
branches of mechanics, it should not be difficult to find 

50 



The Use of Isometric Paper. 




51 



The Use of Isometric Paper. 




52 



The Use of Isometric Paper. 

illustrations of almost anything you wish to make so as 
to see exactly how to go to work if you have any hesitation 
in the matter. It's largely a matter of common sense and 
a little patience, and if you follow the lines, counting the 
spaces to get your various offsets, there will be no diffi- 
culty in handling even complicated drawings in this way. 




53 



The Use of Isometric Paper. 




54 



The Use of Isometric Paper. 




LofC. 



55 



The Use of Isometric Paper. 




56 



Some of the Derry-Collard Co.'s Books. 

Practical Monographs. 

No. I. 

TURNING AND BORING TAPERS. 

Fred H. Colvin. 

A plainh^ written explanation of a subject that puzzles 
many a mechanic. This explains the different ways of des- 
ignating tapers, gives tables, shows how to use the com- 
pound rest and gives the tapers mostly used. (25 Cents.) 

No. 2. 
DRAFTING OF CAMS. 
Louis Rouillion. 

The laying out of cams is a serious problem unless you 
know how to go at it right. This puts you on the right 
road for practically any kind of cam j^ou are likely to run 
up against. And it's plain English, too. (25 Cents.) 

No. 3- 
CO^IMUTATOR CONSTRUCTION. 
\Vm. Baxter, Jr. 

The business end of a dynamo or motor is the com- 
mutator, and this is what is apt to give trouble. This shows 
how they are made, why they get out of whack and what to 
do to put 'em right again. (25 cents.) 

No. 4. 
THREADS AND THREAD CUTTING. 
Colvin-Stabel. 

This clears up many of the mysteries of thread cutting 
such as double and triple threads, internal threads, catching 
threads, use of hobs, etc. Contains a lot of useful hints 
and several tables. (25 Cents.) 

There are others under way — Wiring a House — Brazing and 
Soldering — Inj ectors. 

BEVEL GEAR TABLES. 

A new book that will at once commend itself to mechan- 
ics and draftsmen. Does away with all the trigonometry 
and fancy figuring on bevel gears and makes it easy for any- 
one to lay them out or make them just right. There are 36 
full page tables that show every necessary dimension for all 
sizes or combinations you're apt to need. No puzzling, 

57 



Some of the Derry-Collard Co.'s Books. 

figuring or guessing. Gives placing distance, all the angles 
(including cutting angles) and the correct cutter to use. 
A copy of this prepares you for anything in the bevel gear 
line. 5^x8 inches. (Cloth, $i.oo.) 

"For those having to do with bevel gears this is of decided 
value." — A merican Machinist. 

"Requires no knowledge of trigonometry and will prove emi- 
nently useful." — Iron Trade Review. 



AMERICAN COMPOUND LOCOMOTIVES. 

A new book that shows in a plain, practical way, the 
various features of the compound locomotives in use in 
America. Shows how they are made, what to do when they 
break down or balk. Then there are sections on Valve 
Motion, Piston Valves, Disconnecting, Locating Blows and 
Leaks and Power of Compounds. Includes the newer types 
of tandems and balanced compounds. There are also ten 
special "duotone" inserts on heavy plate paper showing the 
different types. Altogether it's a handsome book as well as 
a practical one. ($1.50.) 

"A practical treatise for practical men." — Railway Age. 
"A singularly practical treatise, copiously illustrated." — Loco- 
motive Magazine, London, 



SWITCHBOARDS. 

Modern electric installations require considerable engi- 
neering in switchboard work and until this was prepared 
there wasn't a book on the subject that was worthy the 
name. This appeals to every engineer or electrician who 
wants to know the practical side of things. It takes up all 
sorts and conditions of dynamos and circuits, shows by dia- 
gram and other illustrations just how it should be done. 
Includes direct, alternating, arc, incandescent and power 
switchboards, as well as the high tension for long distance 
transmission. ($2.00.) 

Not quite ready when this goes to press — consequently no re- 
views to print. 

THE AMERICAN STEEL WORKER. 

The selection, annealing, hardening and tempering of 
steel is a problem that aft'ects practically every manufac- 

58 



Some of the Derry-Collard Co.'s Books. 



turing industry, from making fish hooks to building loco- 
motives. Mr. Markham gives the results of his 27 years 
experience in actual contact with this class of work and 
he is now the acknowledged authority in this country. The 
book is written in plain language that is easily under- 
stood and his methods can be applied to any case. The 
many purchasers express themselves highly pleased with 
its practical value. 343 pages, 5^x8inches (Cloth, $2.50.) 

"Contains over 300 pages of solid instruction." — American 
Blacksmith. 

"Its definite object is to instruct in the selection and treatment 
of steel." — American Machinist. 



BOILER CONSTRUCTION. 

The building of boilers is a work that none have attempt 
ed to describe in detail owing to the necessity of knowing 
each operation thoroughly, in order to do it justice. Here 
is where this book differs from all others. Each step, from 
the first mark on the sheet to the finished boiler, receives 
careful attention in a thoroughly practical way. Locomo- 
tive boilers present more difficulties in laying out and build- 
ing than any other type, and for this reason the author uses 
them as examples. Anyone who can handle them can 
tackle anything. There are over 400 pages and five large 
folding plates. ($3-00.) 

"Gives best methods of largest builders." — Locomotive Engi- 
neering. 

"None fills the same field as this." — American Machinist. 



CHANGE GEAR DEVICES. 

The development of the screw cutting lathe is clearly 
shown in this book from the days of B. C. down to the 
present. Its particular field is the change gear devices that 
have been made and used by various builders. The author 
has sifted the many schemes down to 29, that show som.e 
special points, and these are shown in all the detail that is 
necessary. It is a practical reference book for every ma- 
chine tool builder, designer, draftsman. Printed on enam- 
eled paper; 88 pages; 5^x8 inches. (Cloth, $1.00.) 

"The author has done a service to inventors and designers." — 
Iron Trade Revieiv. 

"Represents in a nutshell a vast amount of work and value to 
designers." — Engineering News. 

59 



JUN 24 1905 










■■:.-v.^l| 




